Thermodynamic Analysis of Ocean Circulation

ThermodynamicAnalysisofOceanCirculationJ.NYCANDER,J.NILSSON,K.DÖÖS,ANDG.BROSTRÖMDepartmentofMeteorology,StockholmUniversity,Stockholm,Sweden(Manuscriptreceived11January2006,infinalform11December2006)ABSTRACTCalculatingastreamfunctionasfunctionofdepthanddensityisproposedasanewwayofanalyzingthethermodynamiccharacteroftheoverturningcirculationintheglobalocean.Thesignofanoverturningcellinthisstreamfunctiondirectlyshowswhetheritisdrivenmechanicallybylarge-scalewindstressorther-mallybyheatconductionandsmall-scalemixing.Itisalsoshownthattheintegralofthisstreamfunctiongivesthethermodynamicworkperformedbythefluid.TheanalysisisalsovalidfortheBoussinesqequations,althoughformallythereisnothermodynamicworkinanincompressiblefluid.Theproposedmethodisappliedbothtoanidealizedcoarse-resolutionthree-dimensionalnumericaloceanmodel,andtotherealistichigh-resolutionOceanCirculationandClimateAdvancedModel(OCCAM).ItisshownthattheoverturningcirculationinOCCAMbetweenthe200-and1000-mdepthisdominatedbyathermallyindirectcellof24Sverdrups(1Sv
106m3s1),forcedbyEkmanpumping.Inthedensestanddeepestwatersthereisathermallydirectcellof18Sv,whichrequiresaforcingbyaround100GWofparameterizedsmall-scalemixing.1.IntroductionIstheoceancirculationthermallyormechanicallyforced?Alreadyacenturyago,Sandström(1908)con-cludedfromhislaboratoryexperimentsthattheheatingandcoolingattheoceansurfacebyitselfwouldnotbeabletoexciteacirculationintheinterioroftheocean.HisargumentswereelaboratedbyJeffreys(1925)andDefant(1961),whoconcludedthatthecirculationmustbemechanicallyforced.Nevertheless,foralongtime,awidespreadviewamongoceanographerswasthatasig-nificantpartoftheoceancirculation(thethermohalinecirculation)isforcedbybuoyancyfluxesatthesurface(mostimportantlybythedifferentialheating).Duringthelastdecadethisissuehasbeendiscussedintensivelyintheoceanographicliterature(MunkandWunsch1998;Huang1999;PaparellaandYoung2002;WunschandFerrari2004;Mullarneyetal.2004;GadeandGustafsson2004;Marchal2007).Wewillnottrytorecapitulatethisdiscussionhere,butmerelypointoutthatithas,inouropinion,sufferedfromalackofad-equateanalysistools.Acommonwayofanalyzingtheenergeticsoftheoceancirculationistoperformbulkcalculationsandconstructboxdiagrams(MunkandWunsch1998;WunschandFerrari2004).Suchadiagramcanbere-gardedasazero-dimensionalpictureoftheocean,withnospatialdegreesoffreedom.Forexample,theglobaltidalforcingisrepresentedbyjustonenumber.How-ever,thesameamountofmechanicalorthermalforc-inghasdifferenteffectsifitisappliedatseveralthou-sandmetersdepthoratthesurface.Therehavealsobeensomeapplicationsofone-dimensionalmodels.OneexampleistheidealizedtubemodeloftheoverturningcirculationproposedbyHuang(1999).AnotheroneisthethermodynamicanalysisbyGadeandGustafsson(2004).Theyplottedahypotheticalclosedparticletrajectory,meanttorepre-senttheoverturningcirculation,inatraditionalther-modynamicpressure–volume(pV)diagram,usingthecorrectequationofstateandrealdataoftheoceanstratification.AsimilarhypotheticalclosedcircuitwasusedinthetheoreticalargumentsbyDefant(1961).Aprincipalproblemwiththeseone-dimensionalmodelsisthattheparticletrajectoriesinathree-dimen-sionalfluidareingeneralnotclosed,eveniftheflowisexactlystationaryandofasimpleandidealizednature.Instead,thetypicalsituationisthatasingletrajectorywilleventuallyfillthewholefluidvolume.ThishasbeenrecognizedforalongtimeinthefieldofnonlinearCorrespondingauthoraddress:J.Nycander,Dept.ofMeteorol-ogy,StockholmUniversity,StockholmS-10691,Sweden.E-mail:[e-mail]2038JOURNALOFPHYSICALOCEANOGRAPHYVOLUME37DOI:10.1175/JPO3113.1©2007AmericanMeteorologicalSocietyJPO3113dynamics(LichtenbergandLieberman1983),andsimi-larfeatureswerealsoseeninarecentstudyofthechaoticnatureofparticletrajectoriesinathree-dimen-sionaloceanmodel(Nycanderetal.2002).Wehereproposeanovelwayofanalyzingtheener-geticsandthermodynamicsoftheoceancirculation:bycomputingthestreamfunctionindepth–densitycoordi-nates.Thiscanbedonebycomputingtheverticaltrans-portacrosslevelsofconstantdepthbetweencurvesofconstantdensity.Alternatively,itcanbedonebycom-putingthetransportacrosssurfacesofconstantdensitybetweenlevelsofconstantdepth.Sincedepthislinearlyrelatedtopressurebythehydrostaticrelation,andden-sityisjusttheinverseofspecificvolume,thisstream-functionineffectdisplaystheglobaloverturningcircu-lationinatraditionalpVdiagram,withimmediatether-modynamicinterpretation.Forexample,thesignofthevariousoverturningcellsdirectlyshowswhethertheyarethermallyormechanicallyforced.(Notethatifsmall-scalemixingisparameterizedasdiffusion,acellthatisforcedbythismixingappearsasthermallyforcedinthemodel.)Thus,thestreamfunctionindepth–densitycoordi-natesgivesatwo-dimensionalthermodynamicpictureoftheoceancirculation,withoutanyassumptionaboutclosedparticletrajectories.However,forthedefinitionofthisstreamfunctiontobeconsistent,thecirculationmustbestationary,atleastinastatisticalsense.Thereareseveralotherwaysofdisplayingtheover-turningcirculationinatwo-dimensionaldiagram.Themostcommononeistoplottheoverturningstream-functionindepth–latitudeco